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Theorem mslb1 25607
Description: The midpoint of a segment AB of the real line is on the "left" of  B. (Contributed by FL, 2-Jan-2008.)
Assertion
Ref Expression
mslb1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  ( ( abs `  ( B  -  A ) )  / 
2 ) )  < 
B )

Proof of Theorem mslb1
StepHypRef Expression
1 2cn 9816 . . . . . 6  |-  2  e.  CC
21a1i 10 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  e.  CC )
3 recn 8827 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
433ad2ant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
5 recn 8827 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  e.  CC )
65, 3anim12i 549 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  e.  CC  /\  A  e.  CC ) )
76ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  e.  CC  /\  A  e.  CC ) )
873adant3 975 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  e.  CC  /\  A  e.  CC ) )
9 abssub 11810 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
108, 9syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( B  -  A ) )  =  ( abs `  ( A  -  B )
) )
1110oveq1d 5873 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( B  -  A )
)  /  2 )  =  ( ( abs `  ( A  -  B
) )  /  2
) )
12 dmse2 25604 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( A  -  B )
)  /  2 )  e.  RR+ )
1311, 12eqeltrd 2357 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( B  -  A )
)  /  2 )  e.  RR+ )
1413rpcnd 10392 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( B  -  A )
)  /  2 )  e.  CC )
152, 4, 14adddid 8859 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( A  +  ( ( abs `  ( B  -  A
) )  /  2
) ) )  =  ( ( 2  x.  A )  +  ( 2  x.  ( ( abs `  ( B  -  A ) )  /  2 ) ) ) )
16 subcl 9051 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B  -  A
)  e.  CC )
178, 16syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
1817abscld 11918 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( B  -  A ) )  e.  RR )
1918recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( B  -  A ) )  e.  CC )
20 2ne0 9829 . . . . . . . 8  |-  2  =/=  0
2120a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  =/=  0 )
2219, 2, 21divcan2d 9538 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( abs `  ( B  -  A ) )  /  2 ) )  =  ( abs `  ( B  -  A )
) )
23 resubcl 9111 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  -  A
)  e.  RR )
2423ancoms 439 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  -  A
)  e.  RR )
25243adant3 975 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR )
26 ltle 8910 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
27263impia 1148 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <_  B )
28 pm3.22 436 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  e.  RR  /\  A  e.  RR ) )
29283adant3 975 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  e.  RR  /\  A  e.  RR ) )
30 subge0 9287 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  ( B  -  A )  <->  A  <_  B ) )
3129, 30syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0  <_  ( B  -  A )  <->  A  <_  B ) )
3227, 31mpbird 223 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  0  <_  ( B  -  A
) )
3325, 32absidd 11905 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( B  -  A ) )  =  ( B  -  A
) )
3422, 33eqtrd 2315 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( abs `  ( B  -  A ) )  /  2 ) )  =  ( B  -  A ) )
3534oveq2d 5874 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  ( 2  x.  ( ( abs `  ( B  -  A
) )  /  2
) ) )  =  ( ( 2  x.  A )  +  ( B  -  A ) ) )
3615, 35eqtrd 2315 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( A  +  ( ( abs `  ( B  -  A
) )  /  2
) ) )  =  ( ( 2  x.  A )  +  ( B  -  A ) ) )
37 2re 9815 . . . . . . . . 9  |-  2  e.  RR
38 remulcl 8822 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
3937, 38mpan 651 . . . . . . . 8  |-  ( A  e.  RR  ->  (
2  x.  A )  e.  RR )
4039adantr 451 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
4140, 24readdcld 8862 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR )
4241recnd 8861 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC )
43 msr4 25606 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) )  e.  RR )
4443recnd 8861 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) )  e.  CC )
451, 20pm3.2i 441 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
46 divmul 9427 . . . . . 6  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC  /\  ( A  +  (
( abs `  ( B  -  A )
)  /  2 ) )  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 ) )  ->  ( (
( ( 2  x.  A )  +  ( B  -  A ) )  /  2 )  =  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) )  <->  ( 2  x.  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) ) )  =  ( ( 2  x.  A
)  +  ( B  -  A ) ) ) )
4745, 46mp3an3 1266 . . . . 5  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC  /\  ( A  +  (
( abs `  ( B  -  A )
)  /  2 ) )  e.  CC )  ->  ( ( ( ( 2  x.  A
)  +  ( B  -  A ) )  /  2 )  =  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) )  <->  ( 2  x.  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) ) )  =  ( ( 2  x.  A
)  +  ( B  -  A ) ) ) )
4842, 44, 47syl2anc 642 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  / 
2 )  =  ( A  +  ( ( abs `  ( B  -  A ) )  /  2 ) )  <-> 
( 2  x.  ( A  +  ( ( abs `  ( B  -  A ) )  / 
2 ) ) )  =  ( ( 2  x.  A )  +  ( B  -  A
) ) ) )
49483adant3 975 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( ( 2  x.  A )  +  ( B  -  A
) )  /  2
)  =  ( A  +  ( ( abs `  ( B  -  A
) )  /  2
) )  <->  ( 2  x.  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) ) )  =  ( ( 2  x.  A
)  +  ( B  -  A ) ) ) )
5036, 49mpbird 223 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  /  2 )  =  ( A  +  ( ( abs `  ( B  -  A )
)  /  2 ) ) )
51 simp3 957 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  B )
52 simp1 955 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
5352recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
54532timesd 9954 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  A )  =  ( A  +  A ) )
5554adantr 451 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( 2  x.  A
)  =  ( A  +  A ) )
5655oveq1d 5873 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( 2  x.  A )  -  A
)  =  ( ( A  +  A )  -  A ) )
57 simpl3 960 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  ->  A  <  B )
585, 5jca 518 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( B  e.  CC  /\  B  e.  CC ) )
59583ad2ant2 977 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  e.  CC  /\  B  e.  CC ) )
6059adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( B  e.  CC  /\  B  e.  CC ) )
61 pncan 9057 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  B )  -  B
)  =  B )
6260, 61syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( B  +  B )  -  B
)  =  B )
6357, 62breqtrrd 4049 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  ->  A  <  ( ( B  +  B )  -  B ) )
643, 3jca 518 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  e.  CC  /\  A  e.  CC ) )
65643ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  e.  CC  /\  A  e.  CC ) )
6665adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( A  e.  CC  /\  A  e.  CC ) )
67 pncan 9057 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( ( A  +  A )  -  A
)  =  A )
6866, 67syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( A  +  A )  -  A
)  =  A )
69 simp2 956 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
7069recnd 8861 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
71702timesd 9954 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  B )  =  ( B  +  B ) )
7271adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( 2  x.  B
)  =  ( B  +  B ) )
7372oveq1d 5873 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( 2  x.  B )  -  B
)  =  ( ( B  +  B )  -  B ) )
7463, 68, 733brtr4d 4053 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( A  +  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )
7556, 74eqbrtrd 4043 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  A  <  B )  -> 
( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )
7651, 75mpdan 649 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  -  A )  <  ( ( 2  x.  B )  -  B ) )
773, 5anim12i 549 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  CC  /\  B  e.  CC ) )
78773adant3 975 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  e.  CC  /\  B  e.  CC ) )
7916ancoms 439 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A
)  e.  CC )
8078, 79syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
8153ad2ant2 977 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
8280, 81negsubd 9163 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  A
)  +  -u B
)  =  ( ( B  -  A )  -  B ) )
8381, 81, 4sub32d 9189 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  B
)  -  A )  =  ( ( B  -  A )  -  B ) )
8453negidd 9147 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  -u A )  =  0 )
855subidd 9145 . . . . . . . . . . . . . . 15  |-  ( B  e.  RR  ->  ( B  -  B )  =  0 )
8685eqcomd 2288 . . . . . . . . . . . . . 14  |-  ( B  e.  RR  ->  0  =  ( B  -  B ) )
87863ad2ant2 977 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  0  =  ( B  -  B ) )
8884, 87eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  -u A )  =  ( B  -  B ) )
89 subcl 9051 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  -  B
)  e.  CC )
9089anidms 626 . . . . . . . . . . . . . . 15  |-  ( B  e.  CC  ->  ( B  -  B )  e.  CC )
9190adantl 452 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  -  B
)  e.  CC )
9278, 91syl 15 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  B )  e.  CC )
93 negcl 9052 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  -u A  e.  CC )
9453, 93syl 15 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  -u A  e.  CC )
9592, 4, 94subaddd 9175 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( B  -  B )  -  A
)  =  -u A  <->  ( A  +  -u A
)  =  ( B  -  B ) ) )
9688, 95mpbird 223 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  B
)  -  A )  =  -u A )
9782, 83, 963eqtr2rd 2322 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  -u A  =  ( ( B  -  A )  + 
-u B ) )
9897oveq2d 5874 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  -u A
)  =  ( ( 2  x.  A )  +  ( ( B  -  A )  + 
-u B ) ) )
99 mulcl 8821 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
1001, 99mpan 651 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
101100adantr 451 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
102 negcl 9052 . . . . . . . . . . . . 13  |-  ( B  e.  CC  ->  -u B  e.  CC )
103102adantl 452 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u B  e.  CC )
104101, 79, 1033jca 1132 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  e.  CC  /\  ( B  -  A
)  e.  CC  /\  -u B  e.  CC ) )
10578, 104syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  e.  CC  /\  ( B  -  A
)  e.  CC  /\  -u B  e.  CC ) )
106 addass 8824 . . . . . . . . . 10  |-  ( ( ( 2  x.  A
)  e.  CC  /\  ( B  -  A
)  e.  CC  /\  -u B  e.  CC )  ->  ( ( ( 2  x.  A )  +  ( B  -  A ) )  + 
-u B )  =  ( ( 2  x.  A )  +  ( ( B  -  A
)  +  -u B
) ) )
107105, 106syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  +  -u B
)  =  ( ( 2  x.  A )  +  ( ( B  -  A )  + 
-u B ) ) )
10898, 107eqtr4d 2318 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  -u A
)  =  ( ( ( 2  x.  A
)  +  ( B  -  A ) )  +  -u B ) )
10953, 100syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  A )  e.  CC )
110109, 4negsubd 9163 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  -u A
)  =  ( ( 2  x.  A )  -  A ) )
111101, 79addcld 8854 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC )
112 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
113111, 112jca 518 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 2  x.  A )  +  ( B  -  A
) )  e.  CC  /\  B  e.  CC ) )
11478, 113syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC  /\  B  e.  CC )
)
115 negsub 9095 . . . . . . . . 9  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  CC  /\  B  e.  CC )  ->  ( ( ( 2  x.  A )  +  ( B  -  A
) )  +  -u B )  =  ( ( ( 2  x.  A )  +  ( B  -  A ) )  -  B ) )
116114, 115syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  +  -u B
)  =  ( ( ( 2  x.  A
)  +  ( B  -  A ) )  -  B ) )
117108, 110, 1163eqtr3d 2323 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  -  A )  =  ( ( ( 2  x.  A )  +  ( B  -  A ) )  -  B ) )
118117breq1d 4033 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B )  <->  ( (
( 2  x.  A
)  +  ( B  -  A ) )  -  B )  < 
( ( 2  x.  B )  -  B
) ) )
119118biimpa 470 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )  ->  ( ( ( 2  x.  A )  +  ( B  -  A ) )  -  B )  <  (
( 2  x.  B
)  -  B ) )
120393ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  A )  e.  RR )
121120, 25readdcld 8862 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  ( B  -  A ) )  e.  RR )
122 remulcl 8822 . . . . . . . . . 10  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
12337, 122mpan 651 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
2  x.  B )  e.  RR )
1241233ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  B )  e.  RR )
125121, 124, 693jca 1132 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR  /\  ( 2  x.  B
)  e.  RR  /\  B  e.  RR )
)
126125adantr 451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )  ->  ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR  /\  ( 2  x.  B )  e.  RR  /\  B  e.  RR ) )
127 ltsub1 9270 . . . . . 6  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR  /\  ( 2  x.  B
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( 2  x.  A )  +  ( B  -  A
) )  <  (
2  x.  B )  <-> 
( ( ( 2  x.  A )  +  ( B  -  A
) )  -  B
)  <  ( (
2  x.  B )  -  B ) ) )
128126, 127syl 15 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )  ->  ( ( ( 2  x.  A )  +  ( B  -  A ) )  < 
( 2  x.  B
)  <->  ( ( ( 2  x.  A )  +  ( B  -  A ) )  -  B )  <  (
( 2  x.  B
)  -  B ) ) )
129119, 128mpbird 223 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( ( 2  x.  A )  -  A
)  <  ( (
2  x.  B )  -  B ) )  ->  ( ( 2  x.  A )  +  ( B  -  A
) )  <  (
2  x.  B ) )
13076, 129mpdan 649 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  A
)  +  ( B  -  A ) )  <  ( 2  x.  B ) )
131 2pos 9828 . . . . . 6  |-  0  <  2
13237, 131pm3.2i 441 . . . . 5  |-  ( 2  e.  RR  /\  0  <  2 )
133 ltdivmul 9628 . . . . 5  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( ( ( 2  x.  A
)  +  ( B  -  A ) )  /  2 )  < 
B  <->  ( ( 2  x.  A )  +  ( B  -  A
) )  <  (
2  x.  B ) ) )
134132, 133mp3an3 1266 . . . 4  |-  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( 2  x.  A )  +  ( B  -  A ) )  / 
2 )  <  B  <->  ( ( 2  x.  A
)  +  ( B  -  A ) )  <  ( 2  x.  B ) ) )
135121, 69, 134syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( ( 2  x.  A )  +  ( B  -  A
) )  /  2
)  <  B  <->  ( (
2  x.  A )  +  ( B  -  A ) )  < 
( 2  x.  B
) ) )
136130, 135mpbird 223 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  A )  +  ( B  -  A ) )  /  2 )  <  B )
13750, 136eqbrtrrd 4045 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  ( ( abs `  ( B  -  A ) )  / 
2 ) )  < 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   RR+crp 10354   abscabs 11719
This theorem is referenced by:  iintlem1  25610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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